Question

If `y=(sinx)^x + sin^-1 sqrtx  "then find"  dy/dx` 

Answer 1

\[\text{ We have, y }= \left( \sin x \right)^x + \sin^{- 1} \sqrt{x}\]

y = u + v

`dy/dx = (du)/dx + (dv)/dx`

`u = (sin x)^x, v = sin^-1 sqrtx`

log u = x log sin x

`1/u (du)/dx = log(sin x) + x 1/sin x cot x`

`du/dx = (sin x)^x [log sin x + x cot x]`

`(dv)/dx = 1/sqrt(1-x) xx 1/(2sqrtx)`

`dy/dx = (sin x)^x [log sin x + x cot x] + 1/(2sqrtx sqrt1-x)`